Mathematics Commons^™

Cultural Traits May Replace Human Mobility Data In Forecasting Covid-19 Mortality: A Deep Learning Approach, Saif Abbas, Tamer Oraby, Michael G. Tyshenko, Samit Bhattacharyya

School of Mathematical & Statistical Sciences Faculty Publications

The COVID-19 pandemic highlighted the need for accurate epidemic forecasting to support public health decision-making. Most existing approaches depend heavily on human mobility data, while largely neglecting population behavior shaped by socio-cultural norms. In this study, we analyze daily COVID-19 mortality and Google mobility data from 72 countries during the first 130 d of the pandemic, a period characterized by high uncertainty and behavioral heterogeneity. In particular, we examine whether Hofstede’s country-level cultural dimensions can serve as latent behavioral forecasters of mortality in lieu of dynamic mobility indicators. Using 100 d for training and 30 d for forecasting, we employ …

The Edge-Distinguishing Game, Nathaniel Benjamin, Elisa Benthem, Cooper Burkel, Marissa E. Chesser, Mike Janssen

Communications on Number Theory and Combinatorial Theory

In this paper, we introduce a graph coloring game called the Edge-Distinguishing Game (EDGe). The edge-distinguishing chromatic number of a graph is used to determine the moves each player can make. We determine which player has a winning strategy for particular graphs and graph families. Additionally, utilizing principles from game theory as well as previous work on a computational solution for the Game of Cycles.

Differentiating And Accommodating Ocd And Anxiety In Mathematical Contexts, Madeline F. Gavares

Interdivisional Studies Summer Fellows

Students in mathematics classrooms may experience emotional and cognitive challenges that affect how they engage with coursework and assessments. While math anxiety has been widely documented, less attention has been given to how obsessive-compulsive disorder (OCD), particularly checking behaviors, may influence students’ experiences in math settings. This study explores how students who report math anxiety and those who report OCD symptoms describe their engagement with math tasks and test-taking. Drawing on survey and interview data from students and teachers, this study examines how these experiences are articulated and perceived in academic math contexts, rather than evaluating ability or outcomes. By …

Faith-Informed Instruction In Secondary Mathematics: A Hermeneutic Phenomenological Study Of Lutheran Church–Missouri Synod Teachers, Jonathan Balsman

Doctoral Dissertations and Projects

The purpose of this hermeneutic phenomenological study was to understand the phenomenon of faith-informed instruction as experienced by secondary mathematics teachers in Lutheran Church–Missouri Synod (LCMS) schools in the United States. The theory guiding this study was Vygotsky’s theory of social constructivism, which frames educators’ instructional and theological meaning-making as socially and contextually developed through collaborative, relational learning environments. The central research question guiding this study was: What are the lived experiences of LCMS secondary mathematics teachers as they practice faith-informed instruction? This study used a hermeneutic phenomenological design to interpret teacher meaning-making and reflective practice. Participants were selected using …

Double-Scoring: Reliable Extraction Of Strong Lottery Tickets, Bryce Christopherson, Jack Baretz, Darian Colgrove, Salah Dandan

Mathematics Faculty Publications

The lottery ticket hypothesis proposes that large random neural networks contain sparse subnetworks that can match the performance of dense models after compara- ble training. A stronger version asserts that sufficiently overparameterized random networks contain subnetworks that are already accurate before any weight training. Existing theory establishes that such strong lottery tickets exist, but reliable ex- traction remains difficult. We revisit edge-popup, a frozen-weight score-training method for extracting strong tickets, and identify layerwise sparsity selection as a central bottleneck. We introduce double-scoring, an augmented score-space parameterization that replaces a layerwise sparsity search with optimization over enlarged score tensors. We prove …

A Bayesian-Optimized Ensemble Deep Learning Framework For Automated Detection And Classification Of Retinal Diseases In Ghana Using Oct Images, Gifty Duah, Eric Nyarko, Gideon Nana Amo, Theophilus Dwamena Frimpong, Anani Lotsi

School of Mathematical & Statistical Sciences Faculty Publications

Retinal diseases pose a significant global health challenge due to their potential to cause severe visual impairment and blindness. This study aimed to develop a robust deep learning ensemble framework for the automated detection and classification of retinal diseases from optical coherence tomography (OCT) images. This study used OCT images from WATBORG Eye Services in Ghana, including glaucoma, macular edema, posterior vitreous detachment (PVD), and healthy eyes. The data preprocessing steps included augmentation, resizing, and one-hot encoding. The dataset was divided into training (56%), validation (14%), and testing (30%) sets using stratified sampling. Six convolutional neural network (CNN) architectures, Visual …

Biharmonic Curves In Warped Product Manifolds I ×FMN(C), Şaban Güvenç, Ci̇han Özgür

Turkish Journal of Mathematics

We explore the geometric properties of biharmonic curves in warped product manifolds of the form I ×_f Mⁿ(c), where I is an open interval and Mⁿ(c) is a space of constant curvature. By establishing a main theorem, we analyze four distinct cases to reveal deeper curvature-related characteristics of these curves, including situations where they are slant. Finally, we construct three examples in I ×_f S²(1).

Generalized Submaximality And Paracompactness In Ideal Topological Spaces Via Δβ*-𝔍-Open Sets, Chawalit Boonpok, Areeyuth Sama-Ae

Turkish Journal of Mathematics

This paper explores the notions of δβ^*-𝔍-submaximality and δβ^*-𝔍-paracompactness within ideal topological spaces, viewing them as natural generalizations of the traditional concepts of submaximality and paracompactness. These concepts arise naturally in the context of ideal topology, where an ideal 𝔍 on a topological space provides additional structure for analyzing topological properties. It focuses on the study of submaximal spaces by employing δβ^*-𝔍-open sets, which serve as fundamental building blocks for understanding the topological behavior in ideal spaces. Furthermore, the work provides multiple characterizations of δβ^*-𝔍-paracompact spaces and examines how this property behaves …

Notes On The Geometry Of Electromagnetic Fields And Maxwell’S Equations Along A Nonnull Curves In Nonflat-3d Space Forms MQ3(C), Fatma Almaz, Cumali̇ Eki̇ci̇

Turkish Journal of Mathematics

In this paper, the directional derivatives in accordance with the orthonormal frame {T^→, N^→, B^→} are defined in M_q³(c), the extended Serret-Frenet relations by using Frenet formulas are expressed. Furthermore, we express the bending elastic energy function for the same particle in M_q³(c) according to curve γ(s, ξ, η) and we tried to interpret it from a geometric perspective of the energy for unit vector fields. Also, Maxwell’s equations are expressed for the electric and magnetic field vectors …

Propagation Of Dirac Spherical Waves In The Expanding Universe, Karen Yagdjian

School of Mathematical & Statistical Sciences Faculty Publications

The explicit formulas for the spherical solutions of the Dirac equation in the expanding universe are given. The initial value of the solution can be, in particular, a wave function of the hydrogen-like atom or a spherical wave in the Minkowski space, that then propagates in the Friedmann-Lemaître-Robertson-Walker space-time, which is expanding with the de Sitter scale factor.

Observations On Recurrent Loss In The Neural Network Model Of A Partial Differential Equation: The Advection–Diffusion Equation, Jonah A. Reeger

Faculty Publications

A growing body of literature has been leveraging techniques of machine learning (ML) to build novel approaches to approximating the solutions to partial differential equations. Noticeably absent from the literature is a systematic exploration of the stability of the solutions generated by these ML approaches. Here, a recurrent network is introduced that matches precisely the evaluation of a multi-step method paired with a collocation method for approximating spatial derivatives in the advection–diffusion equation. This allows for two things: (1) the use of traditional tools for analyzing the stability of a numerical method for solving PDEs and (2) bringing to bear …

Comparative Analysis Of Traditional And Deep Learning Time Series Architectures For Influenza A Infectious Disease Forecasting, Edmund Fosu Agyemang, Hansapani Rodrigo, Vincent Agbenyeavu

School of Mathematical & Statistical Sciences Faculty Publications

Influenza A remains a major cause of respiratory mortality worldwide, motivating accurate forecasting to support timely preparedness and resource allocation. This study presents a comparative evaluation of two traditional seasonal time series baselines, ARIMA and Holt–Winters exponential smoothing (ETS), and six deep learning (DL) architectures (Simple RNN, LSTM, GRU, BiLSTM, BiGRU, and a Transformer) for forecasting monthly Influenza A case counts in the United States. Data from January 2009 to December 2023 were analyzed, using January 2009 to December 2022 for training and January 2023 to December 2023 for out-of-sample testing. Models were tuned using a validation split and assessed …

Assessing The Role Of Model Complexity In Virtual Clinical Trial Outcomes, Jana L. Gevertz, Joanna R. Wares

Department of Math & Statistics Faculty Publications

Virtual clinical trials (VCTs) hold significant promise for improving the drug development process, yet their predictive reliability depends critically on design decisions that remain poorly understood. This study examines how model complexity influences VCT outcomes, as well as how the choice of prior parameter distributions and virtual patient inclusion criteria affects those outcomes. Using oncolytic virotherapy treatment of murine tumors as a case study, we compared a relative hierarchy of three mathematical models of increasing complexity under different parameter priors (uniform and normal distributions) and two inclusion methods (accept-or-reject and accept-or-perturb). Our results demonstrate that the simplest model produces a …

Operator Theoretic Methods For Coupled Pde In General Geometries, Yuhao Mu

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

This thesis develops and applies operator theoretic methods applied to coupled PDE analysis in general geometries, in which the coupling involves interchange across a boundary. The general geometries refer to domains with merely Lipschitz continuous boundaries (e.g. any non-convex polygon), as opposed to those with any further smoothness conditions on the boundary. The key results include a new pressure elimination method for coupled fluid–structure interaction (FSI) PDEs with boundary interchange, which allows an explicit semigroup representation of the PDE in terms of just the fluid and structure variables. This novel technique, valid over arbitrary bounded Lipschitz domains, leads to a …

On Universal C∗ Algebras Associated To Operator Spaces And Generalized Crossed Products, Sayan Kansa Banik

Doctoral Theses

In my thesis, we study generalized crossed product constructions of the group \( C^* \)-algebra \( C^*(G) \) with respect to certain completely positive maps, where \( G \) is assumed to be a discrete amenable group. We also investigate the universal \( C^* \)-algebra \( \mathcal{E}_\alpha \) introduced by Hirshberg, which is constructed from \( C^*(G) \) and a pure injective homomorphism \( \alpha \colon G \to G \). In particular, we analyze its relationship with Exel's construction of generalized crossed products associated with the endomorphism of \( C^*(G) \) induced by \( \alpha \), together with an appropriate …

Pseudodifferential Absorbing Boundary Conditions For Waves, Lauren Taylor

Mechanical Engineering Theses

Absorbing boundary conditions (ABCs) are required to truncate the computational domain when performing Finite Element Analyses of exterior acoustic problems where physical domain is unbounded. ABC is applied on a fictitious boundary containing the scatterer and ideally allows outgoing waves to leave without non-physical reflections. Preventing artificial reflections is essential to benefit from the accuracy of the numerical method used. Otherwise, ABC acts as a reflective surface, and artificial reflections distort the solution in the entire domain which is not recoverable by any type of refinement. Pseudodifferential ABCs were used to describe the Dirichlet-to-Neumann map which maps known boundary values …

Analytic Solutions To Oceanographic Kinematic Property Equations, James Turbett

Theses and Dissertations

With the recent advent of large submesoscale drifter deployments, interpreting the timeseries of kinematic properties (KP) - divergence, vorticity, shear, and normal strain rates observed therein is a pressing issue. The evolution equations for the four KP are derived from the two-dimensional momentum conservation equations. The resulting equations, a nonlinear coupled system of ordinary differential equations, capture the submesoscale motion of drifters along fixed ocean surfaces, with time and space scales on the order of days and kilometers. This thesis builds theory around KP timeseries behavior by solving for the analytic solutions in the particular cases that one KP is …

Open Neighbourhood Edge Degree And Metric Descriptors Qspr And Multi Criteria Analysis Of Chemicals And Antiviral Drugs, Gayathri A Ms

Theses and Dissertations

Chemical Graph theory offers powerful computational methods for evaluating chemical structures, significantly impacting cheminformatics and drug analysis. It provides a robust mathematical framework to represent, model, and analyse molecular systems through topological indices and graph invariants. These indices serve as effective numerical descriptors, encapsulating significant structural information that enhances the prediction of physicochemical and biological properties. This thesis utilizes mathematical modeling, algorithmic computation, and decision making process to derive extensive insights to drug discovery and molecular analysis.

The thesis introduces a new class of topological indices ONE1, ONE2, ONE3, ONE4, ONE5,ONE6 and ONE7 based on open-neighbourhood-edge-degree and it evaluates the …

Coupled-Pendulum Modeling In An Ode Class: An Assignment On Fourier-Initialized Gradient Descent In Machine Learning, Huy Truong, Andrew Bennett

CODEE Journal

As data-driven methods are increasingly used in science and engineering, students benefit from learning to integrate machine learning techniques with traditional mathematical modeling. We present a hands-on extra-credit assignment for an undergraduate ordinary differential equations (ODE) course that enables students to compare classical analytical methods with data-driven approaches on the same physical system. Using a coupled-pendulum system---two pendulums connected by a spring---with real experimental data acquired via video tracking of a real physical setup, students work through three models in a guided Jupyter notebook with all code provided. First, they fit a neural network with Fourier features as a purely …

Differential Topology And The Poincaré-Hopf Theorem, Tara Saini

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we will develop the ideas needed to understand and prove the Poincaré–Hopf Theorem, which connects the local behavior of smooth vector fields to global topological properties. We will begin by introducing smooth manifolds and smooth maps, which are the basis of differential topology. We will then define derivatives of smooth maps through tangent spaces and use these to classify points. To build toward the theorem, we will introduce orientation, degree, and smooth vector fields. These concepts will culminate in a proof of the Poincaré–Hopf Theorem, aided by Brouwer’s Fixed Point Theorem. Finally, we will apply the result …